MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itg2val Unicode version

Theorem itg2val 19099
Description: Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypothesis
Ref Expression
itg2val.1  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
Assertion
Ref Expression
itg2val  |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  ) )
Distinct variable group:    x, g, F
Allowed substitution hints:    L( x, g)

Proof of Theorem itg2val
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 xrltso 10491 . . 3  |-  <  Or  RR*
21supex 7230 . 2  |-  sup ( L ,  RR* ,  <  )  e.  _V
3 reex 8844 . 2  |-  RR  e.  _V
4 ovex 5899 . 2  |-  ( 0 [,]  +oo )  e.  _V
5 breq2 4043 . . . . . . 7  |-  ( f  =  F  ->  (
g  o R  <_ 
f  <->  g  o R  <_  F ) )
65anbi1d 685 . . . . . 6  |-  ( f  =  F  ->  (
( g  o R  <_  f  /\  x  =  ( S.1 `  g
) )  <->  ( g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) ) )
76rexbidv 2577 . . . . 5  |-  ( f  =  F  ->  ( E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) )  <->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) ) )
87abbidv 2410 . . . 4  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) } )
9 itg2val.1 . . . 4  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
108, 9syl6eqr 2346 . . 3  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  L )
1110supeq1d 7215 . 2  |-  ( f  =  F  ->  sup ( { x  |  E. g  e.  dom  S.1 (
g  o R  <_ 
f  /\  x  =  ( S.1 `  g ) ) } ,  RR* ,  <  )  =  sup ( L ,  RR* ,  <  ) )
12 df-itg2 18993 . 2  |-  S.2  =  ( f  e.  ( ( 0 [,]  +oo )  ^m  RR )  |->  sup ( { x  |  E. g  e.  dom  S.1 ( g  o R  <_  f  /\  x  =  ( S.1 `  g
) ) } ,  RR* ,  <  ) )
132, 3, 4, 11, 12fvmptmap 6820 1  |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632   {cab 2282   E.wrex 2557   class class class wbr 4039   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Rcofr 6093   supcsup 7209   RRcr 8752   0cc0 8753    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884   [,]cicc 10675   S.1citg1 18986   S.2citg2 18987
This theorem is referenced by:  itg2cl  19103  itg2ub  19104  itg2leub  19105  itg2addnclem  25003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-itg2 18993
  Copyright terms: Public domain W3C validator